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標題: Tailored finite point methods and radial basis function collocation methods for solving partial differential equations and their applications
作者: 蔡至清
Chih-Ching Tsai
引用: [1] N. Akhmediev and A. Ankiewice, Partially coherent solitons on a finite back- ground, Phys. Rev. Lett., 82 (1999) 2661-2664. [2] H. Arkin, L.X. Xu and K.R. Holmes, Recent developments in modeling heat transfer in blood perfused tissues, IEEE Trans. Biomed. Eng., 41 (1994) 97- 107. [3] E.L. Allgower and K. Georg, An introduction to numerical continuation meth- ods, SIAM Publications, Philadelphia, 2003. [4] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995) 198. [5] A. Avila and L. Jeanjean, A result on singularly perturbed elliptic problems, Comm. Pure App. Anal., 4.2 (2005), 343-358. [6] I. Babu˘ka and J.E. Osborn, Eigenvalue Problems, Handbook of Numericals Analysis, Finite Element Methods (Part 1) (P.G. Ciarlet and J.L. Lious, Eds.), Vol. 2, 641-787, Amsterdam, 1991. [7] L.O. Baksmaty, Y. Liu, U. Landmanc, N.P. Bigelowd and H. Pu, Numeri- cal exploration of vortex matter in Bose-Einstein condensates, Math. Comput. Simulat., 80 (2009) 131-138. [8] A. Banerjee, A. Ogale, C. Das, K. Mitra and C. Subramanian, Temperature distribution in different materials due to short pulse laser irradiation, Heat Transfer Eng., 26 (2005) 41-49. [9] W. Bao and H. Wang, Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional, J. Comput. Phys., 187 (2003) 230- 254. [10] W. Bao and H. Wang, An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates, J. Comput. Phys., 217 (2006) 612-626. [11] V. Bayona, M. Moscoso, M. Kindelan, Optimal constant shape parameter for multiquadric based RBF-FD method, J. Comput. Phys., 230 (2011) 7384-7399. [12] S.N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift fur Physik, 26 (1924) 178-181. [13] D.A. Butts and D.S. Rokhsar, Predicted signatures of rotating Bose-Einstein condensates, Nature, 397 (1999) 327. [14] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schr dinger equationo with a general nonlinearity, Arch. Rational Mech. Anal., 185 (2007), 185-200. [15] S.-L. Chang and C.-S. Chien, Adaptive continuation algorithms for computing energy levels of rotating Bose-Einstein condensate, Comput. Phys. Commun., 177 (2007) 707-719. [16] S.-L. Chang, C.-S. Chien and Z.-C. Li, A finite difference continuation method for computing energy levels of Bose-Einstein condensates, Comput. Phys. Com- mun., 179 (2008) 208-226. [17] H.-S, Chen, S.-L. Chang and C.-S. Chien, Spectral collocation methods using sine functions for a rotating Bose-Einstein condensation in optical lattices, J. Comput. Phys., 231 (2012) 1553-1569. [18] J.-S.Chen, L. Wang, H.-Y. Hu and S.-W.Chi, Subdomain radial basis collocation method for heterogeneous media, Int. J. Numer. Meth. Engng., 80 (2009) 163- 190. [19] A.H.-D. Cheng, M.A. Golberg, E.J. Kansa and G. Zammito, Exponential con- vergence and h-c multiquadric collocation method for partial differential equa- tions, Numer. Meth. Part. Diff. Eq., 19 (2003) 571-594. [20] J.Y. Choo and D.H. Schultz, High order methods for differential equations with small coefficients for the second order terms, Comput. Math. Appl., 25 (1993) 105-123. [21] I. Danaila and F. Hecht, A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates, J. Comput. Phys., 229 (2010) 6946-6960. [22] K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995) 3969-3973. [23] M.W. Dewhirst, B.L. Viglianti, M. Lora-Michiels, M. Hanson and P.J. Hoopes, Basic principles of thermal dosimetry and thermal thresholds for tissue damage from hyperthermia, Int. J. Hyperthermia, 19 (2003) 267-294. [24] I.S. Duff, A.M. Erisman and J.K. Reid, Direct Methods for Sparse Matrices, Math. Comput., 52 (1989) 250-252. [25] A. Einstein, Quantentheorie des einatomigen idealen Gases. Zweite Abhand- lung, Sitzber. Preuss. Akad. Wiss., (1925) 3-10. [26] A. Fasano, D. Homberg and D. Naumov, On a mathematical model for laser- induced thermotherapy, Math. Comput. Model., 34 (2010) 3831-3840. [27] G.E. Fasshauer, Solving partial differential equations by collocation with radial basis functions, (A. LeM´haut´, C. Rabut, L.L. Schumaker, Eds.), Chamonixee Proceedings, Vanderbilt University Press, Nashville, TN, 1996. [28] G.E. Fasshauer, Solving differential equations with radial basis functions: mul- tilevel methods and smoothing, Adv. Comput. Math., 11 (1999) 139-159. [29] G.E. Fasshauer, Newton iteration with multiquadrics for the solution of nonlin- ear PDEs, Comput. Math. Appl., 43 (2002) 423-438. [30] Y. Feng and D. Fuentes, Model-based planning and real-time predictive control for laser induced thermal therapy, Int. J. Hyperthermia, 27 (2011) 751-761. [31] A.J.M. Ferreira, C.M.C. Roque, R.M.N. Jorge and E.J. Kansa, Static deforma- tions and vibration analysis of composite and sandwich plates using a layer-wise theory and multiquadrics discretization, Eng. Anal. Bound. Elem., 29 (2005) 1104-1114. [32] A.L. Fetter, A.A. Svidzinski, Vortices in a trapped dilute Bose-Einstein con- densate, J. Phys. Condens. Matter, 13 (2001) R135. [33] A.L. Fetter, B. Jackson and S. Stringari, Rapid rotation of a Bose-Einstein condensate in a harmonic plus quartic trap , Phys. Rev. A, 71 (2005) 013605. [34] B. Fornberg and C. Piret, On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere, J. Comput. Phys., 227 (2008) 2758-2780. [35] R. Franke, Scattered data interpolation: Tests of some methods, Math. Comp., 38 (1982) 181-200. [36] C. Franke and R. Schaback, Convergence order estimates of meshless collocation methods using radial basis functions. Adv. Comput. Math., 8 (1998) 381-399. [37] C.-T. Germer, C.M. Isbert, D. Albrecht, J. P. Ritz, A. Schilling, A. Roggan, K.-J. Wolf, G. M ller and H.-J. Buhr, Laser-induced thermotherapy for theu treatment of liver metastasis, Surg. Endosc., 12 (1998) 1317-1325. [38] J. Goldak, A. Chakravarti and M. Bibby, A new finite element model for welding heat sources, Metall Trans B, 15B (1984) 299-305. [39] L. Goldman, R. Wilson, P. Hornby and R. Meyer, Biomedical aspects of the laser, Springer Science Business Media. LLC, 1967. [40] G.H. Golub and C. F. van Loan, Matrix computations, 3rd Ed., Johns Hopkins University Press, 1996. [41] G.H. Golub and Q. Ye, Inexact inverse iteration for generalized eigenvalue problems, BIT, 40 (2000) 671-684. [42] H. Grad, Magnetofluid-dynamic spectrum and low shear stability, Proc. Nat. Acad. Sci. USA Part I, 70 (1973) 12, 3277-3281. [43] M. Greiner, O. Mandel, T. Esslinger, T.W. Hnsch and I. Bloch, Quantume phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature, 451 (2002) 39-44. [44] A. Griffin, T. Nikuni and E. Zaremga, Bose-condensed gases at finite temper- atures, Cambridge University Press, Cambridge, 2009. [45] P.P.N. de Groen and M. van Veldhuizen, A stabilized Galerkin method for convection-diffusion problems, SIAM J. Sci. Statist. Comput., 10 (1989) 274- 297. [46] E.P. Gross, Structure of a quantized vortex in boson systems, Nuovo. Cimento., 20 (1961) 454-477. [47] R.W.Y. Habash, R. Bansal, D. Krewski and H.T. Alhafid, Thermal Therapy, Part 1: An introduction to thermal therapy, Crit. Rev. Biomed. Eng., 34 (2006) 459-489. [48] H. Han and Z. Huang, A tailored finite point method for the Helmholtz equation with high wave numbers in heterogeneous medium, J. Comp. Math., 26 (2008) 728-739. [49] H. Han and Z. Huang, Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions, J. Sci. Comp., 41 (2009) 200-220. [50] H. Han and Z. Huang, Tailored finite point method for steady-state reaction- diffusion equations, Commun. Math. Sci., 8 (2010) 887-899. [51] H. Han and Z. Huang, Tailored finite point method based on exponential bases for convection-diffusion-reaction equation, Math. Comp., 82 (2012) 213-226. [52] H. Han, Z. Huang and R.B. Kellogg, A tailored finite point method for a singular perturbation problem on an unbounded domain, J. Sci. Comp., 36 (2008) 243- 261. [53] H. Han, J.J.H. Miller and M. Tang, A parameter-uniform tailored finite point method for singularly perturbed linear ODE systems, J. Comp. Math., 31 (2013) 422-438. [54] H. Han, M. Tang, and W. Ying, Two uniform tailored finite point schemes for the two dimensional discrete ordinates transport equations with boundary and interface layers, Commun. Comput. Phys., 15 (2014) 797-826. [55] H. Han and Z. Zhang, Multiscale tailored finite point method for second order elliptic equations with rough or highly oscillatory coefficients, Commun. Math., Sci. 10 (2012) 945-976. [56] R.L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 176 (1971) 1905-1915. [57] W.A. Harris Jr., Singular perturbations of eigenvalue problems, Arch. Ration. Mech. Anal., 7 (1961) 224-241. [58] M. R. Hestenes and E. Steifel, Methods of conjugate gradient for solving linear systems, J. of Res. Nat. Bureau Standards, 49 (1952) 409-436. [59] P. Hsieh, Y. Shih and S. Yang, A tailored finite point method for solving steady MHD duct flow problems with boundary layers, Commun. Comput. Phys., 10 (2011) 161-182. [60] H.-Y. Hu, Z.-C. Li and A.H.-D. Cheng, Radial basis collocation methods for elliptic boundary value problems, Comput. Math. Appl., 50 (2005) 289-320. [61] C.S. Huang, C.F. Lee and A.H.D. Cheng, Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method, Eng. Anal. Bound. Elem., 31 (2007) 614-623. [62] Z. Huang and X Yang, Tailored finite point method for first order wave equation, J. Sci. Comput., 49 (2011) 351-366. [63] E.M. de Jager and T. K pper, The Schrdinger equation as a singularly per-uo turbation problem, Proc. of the Royal Soc. of Edinburgh, 82A (1978) 1-11. [64] S.L. Jacques, Lase-tissue interactions: photochemical, photothermal, and pho- tomechanical, Surg. Clin. N. Am., 72 (1992) 531-558. [65] M. Jaunich, S. Raje, K. Kim, K. Mitra and Z. Guo, Bio-heat transfer analysis during short pulse laser irradiation of tissues, Int. J. Heat Mass Transf., 51 (2008) 5511-5521. [66] B.-W.Jeng,Y.-S.Wang,C.-S.Chien, A two-parameter continuation algorithm for vortex-pinning of rotating BEC, Comput. Phys. Commun., 184 (2013) 493-508. [67] E.J. Kansa, Multiquadrics - a scattered data approximation scheme with appli- cations to computational fluid dynamics-I., Comput. Math. Appl., 19 (1990) 127-145. [68] K. Kasamatsu and M. Tsubota,Dynamical properties of vortices in a Bose Ein- stein condensate in a rotating lattice, J. Low Temp. Phys., 148 (2007) 357. [69] K. Kasamatsu, M. Tsubota and M. Ueda, Giant hole and circular superflow in a fast rotating Bose-Einstein condensate, Phys. Rev. A, 66 (2002) 053606. [70] C.T. Kelly, Iterative methods for linear and nonlinear equations, SIAM Philadelphia, 1995. [71] M. Kindelan, F. Bernal, P. P. Gonzalez-Rodriguez, M. Moscoso, Application of the RBF meshless method to the solution of the radiative transport equation, J. Comput. Phys., 229 (2010) 1897-1908. [72] Y.-C. Kuo,W.-W. Lin,S.-F. Shieh and W. Wang, Exploring bistability in rotating Bose-Einstein condensates by a quotient transformation invariant continuation method, Physica D, 240 (2011) 78-88. [73] A.N. Krylov, On the numerical solution of the equation by which the frequency of small oscillations is determined in technical problems, Izv. Akad. Nauk SSSR Ser. Fiz.-Mat., 4 (1931) 491-539. [74] Y.-L. Lai, K.-Y. Lin, and L. Wen-Wei, An inexact inverse iteration for large sparse eigenvalue problems, Numer. Linear Algebra Appl., 1 (1997) 1-13. [75] C. Lanczos, Solution of systems of linear equations by minimized iterations, Journal of Research of the National Bureau of Standards J. Res. Nat. Bur. Stand., 49 (1952) 33-53. [76] L.D. Landau and E.M. Lifshitz, Quantum mechanics, non-relativistic theory, Pergamon Press, 1977. [77] I.V. Larina, K.V. Larin and R.O. Esenaliev, Real-time optoacoustic monitoring of temperature in tissues, J. Phys. D: Appl. Phys., 38 (2005) 2633-2639. [78] S. Leble and B. Reichel, Coupled nonlinear Schrdinger equations in optic fiberso theory - From general to solitonic aspects, Eur. Phys. J. Special Topics, 173 (2009) 5-55. [79] G.L. LeCarpentier, M. Motamedi, L.P. McMath, S. Rastegar and A.J. Welch, Continuous wave laser ablation of tissue: analysis of thermal and mechanical events, IEEE Trans. Biomed. Eng., 40 (1993) 188-200. [80] C.K. Lee, X. Liu and S.C. Fan, Local multiquadric approximation for solving boundary value problems, Comput. Mech., 30 (2003) 396-409. [81] A.D. Lercher, T. Takekoshi, M. Debatin, B. Schuster, R. Rameshan, F. Fer- laino, R. Grimm and H.-C. Ngerl, Production of a dual-species Bose-Einsteina condensate of Rb and Cs atoms, Eur. Phys. J. D, 65 (2011) 3-9. [82] K. Li , Q.B. Huang, J.L. Wang, L.G. Lin, An improved localized radial basis function meshless method for computational aeroacoustics, Eng. Anal. Bound. Elem., 35 (2011) 47-55. [83] Z.-C. Li, S.-Y. Chen, C.-S. Chien and H.-S. Chen, A spectral collocation method for a rotating Bose-Einstein condensation in optical lattices, Comput. Phys. Commun., 182 (2011) 1215-1234. [84] L. Ling and E.J. Kansa, A least-squares preconditioner for radial basis functions collocation methods, Adv. Comput. Math., 23 (2005) 31-54. [85] J. Liu, X Chen and L.X. Xu, New thermal wave aspects on burn evaluation of skin subjected to instantaneous heating, IEEE Trans. Biomed. Eng., 46 (1999) 420-428. [86] K.W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate, Phys. Rev. Lett., 84 (2000) 806. [87] W.R. Madych, Miscellaneous error bounds for multiquadric and related inter- polations, Comput. Math. Appl., 24 (1992) 121-138. [88] M.R. Matthews, B.P. Anderson, P.C. Halian, D.S. Hall, C.E. Wieman and E.A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett., 83 (1999) 2498. [89] C. Micchelli, Interpolation of scattered data: Distance matrices and condition- ally positive definite functions, Constr. Approx., 2 (1986) 11-22. [90] M. Mu, L. Sun and H.A. Dijkstra, The sensitivity and the stability of the ocean's thermohaline circulation to finite amplitude perturbations, J. Phys. Oceanogr., 34 (2004) 2305-2315. [91] A. Narasimhan and S. Sadasivam, Non-Fourier bio heat transfer modelling of thermal damage during retinal laser irradiation, Int. J. Heat Mass Transf., 60 (2013) 591-597. [92] J.T. Oden, K.R. Diller, C. Bajaj, J.C. Browne, J. Hazle, I. Babu˘ka, J. Bass, L.s Biduat, L. Demkowic, A. Elliott, Y. Feng, D. Fuentes, S. Prudhomme, M.N. Ry- lander, R.J. Stafford and Y. Zhang, Dynamic data-driven finite element models for laser treatment of cancer, Numer. Methods Partial Differ. Equ., 23 (2007) 904-922. [93] E. O˜ ate, S. Idelsohn, O.C. Zienkiewicz and R.L. Taylor, A finite point methodn in computational mechanics. Applications to convective transport and fluid flow, Int. J. Numer. Methods Eng., 39 (1996) 3839-3866. [94] J. Overgaard, D. Gonzalez Gonzalez, M.C.C.H. Hulshof, G. Arcangeli, O. Dahl, O. Mella and S.M. Bentzen, Hyperthermia as an adjuvant to radiation therapy of recurrent ormetastatic melanoma: a multicenter randomized trial by the Eu- ropean society for hyperthermic oncology, Int. J. Hyperthermia, 12 (1996) 3-20. [95] B.N. Parlett, The Symmetric Eigenvalue Problem, Prentic-Hall, Englewood Cliffs, NJ, 1980. [96] V. Pavelic, R. Tanbakuchi, O.A. Uyehara and P.S. Myers, Experimental and computed temperature histories in gas tungsten arc welding of thin plates, Weld. J., 48 (1969) 295s-305s. [97] A.A. Penckwitt, R.J. Ballagh and C.W. Gardiner, Nucleation, growth, and stabilization of Bose-Einstein condensate vortex lattices, Phys. Rev. Lett., 89 (2002) 260402. [98] H.H. Pennes, Analysis of tissue and arterial blood temperatures in the resting forearm, J. Appl. Physiol. 1 (1948) 93-122. [99] L.P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP., 13 (1961) 451-454. [100] H. Power and V. Barraco, A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations, Comput. Math. Appl., 43 (2002) 551-583. [101] H. Pu, L.O. Baksmaty, S. Yi and N.P. Bigelow, Structural phase transitions of vortex matter in an optical lattice, Phys. Rev. Lett., 94 (2005) 190401. [102] C. Raman, J. R. Abo-Shaeer, J.M. Vogels, K. Xu and W. Ketterle, Vortex nucleation in a stirred Bose-Einstein condensates, Phys. Rev. Lett., 87 (2001) 210402. [103] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Anal- ysis of Operators, Academic Press, Inc., 1978. [104] C.M.C. Roque, J.D. Rodrigues and A.J.M. Ferreira, Analysis of thick plates by local radial basis functions-finite differences method, Meccanica, 47 (2012) 1157-1171. [105] P. Rosenbuch, V. Bretin and J. Dalibard, Three-dimensional vortex config- urations in a rotating Bose-Einstein condensate, Phys. Rev. Lett., 89 (2002) 200403. [106] H.G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly per- turbed differential equations: convection-diffusion and flow problems, Springer, 1996. [107] Y. Saad, Iterative methods for sparse linear systems, 2nd Ed., SIAM Publica- tions, Philadelphia, 2003. [108] Y. Saad and M. Schultz, GMRES a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986) 856-869. [109] S.A. Sapareto and W.C. Dewey, Thermal dose determination in cancer ther- apy, Int. J. Radiat. Oncol. Biol. Phys., 10 (1984) 787-800. [110] B. Sarler, R. Vertnik, Meshfree explicit local radial basis function collocation method for diffusion problems, Comput. Math. Appl., 21 (2006) 1269-1282. [111] C. Schmeiser and R. Weiss, Asymptotic analysis of singularly perturbed bound- ary value problems, SIAM J. Math. Anal., 17 (1986) 560-579. [112] K. Schulten, Notes on quantum mechanics, Department of Physics and Beck- man Institute, University of Illinois at Urbana-Champaign, 1999. [113] R. Shankar, Principles of quantum mechanics, 2nd Ed., New York, Kluwer Academic/Plenum Publishers, 1994. [114] W. Shen and J. Zhang, Modeling and numerical simulation of bioheat transfer and biomechanics in soft tissue, Math. Comput. Model., 41 (2005) 1251-1265. [115] S.-D. Shih and R.B. Kellogg, Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18 (1987) 1647-1511. [116] Y. Shih, R.B. Kellogg and Y. Chang, Characteristic tailored finite point method for convection-dominated convection-diffusion-reaction problems, J. Sci. Com- put., 47 (2011) 198-215. [117] Y. Shih, R.B. Kellogg and P. Tsai, A Tailored finite point method for convection-diffusion-reaction problems, J. Sci. Compu., 43 (2010) 239-260. [118] A.D. Snider, Partial differential equations: sources and solutions, Prentice Hall, 1999. [119] D. Stevens, H. Power, M. Lees and H. Morvan, The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems, J. Comput. Phys., 228 (2009) 4606-4624. [120] S. Stock, V. Bretin, F. Chevy and J. Dalibard, Shape oscillation of a rotating Bose-Einstein condensate, Europhys. Lett., 65 (2004) 594-600. [121] C. Sturesson and S. Andersson-Engels, A mathematical model for predicting the temperature distribution in laser-induced hyperthermia. Experimental eval- uation and applications, Phys. Med. Biol., 40 (1995) 2037-2052. [122] H.D. Suit and L.E. Gerweck, Potential for hyperthermia and radiation therapy, Cancer Res., 39 (1979) 2290-2298. [123] D.B. Szyld, Criteria for combining inverse and Rayleigh quotient iteration, SIAM J. Numer. Anal., 53 (1988) 1369-1375. [124] K. Ting, K.-T. Chen, S.-F. Cheng, W.-S. Lin, and C.-R. Chang, Prediction of skin temperature distribution in cosmetic laser surgery, Jpn. J. Appl. Phys., 47 (2008) 361-367. [125] K. Ting, K.T. Chen, Y.L. Su and C.J. Chang, Effects of thermal energy ac- cumulations of multi-point heat sources in laser cosmetic surgery, Proc. Inst. Mech. Eng. Part E-J. Process Mech. Eng., 228 (2014) 162-167. [126] D.A. Torvi and J.D. Dale, A finite element model of skin subjected to a flash fire, ASME J. Biomech. Eng., 116 (1994) 250-255. [127] H. A. Van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant to Bi-CG for the solution of nonsymmetric systems, SIAM J. Sci. Statist. Comput., 13 (1992), 631-644. [128] L. Wang, J.-S. Chen and H.-Y. Hu, Subdomain radial basis collocation method for fracture mechanics, Int. J. Numer. Meth. Engng., 83 (2010) 851-876. [129] Y.-S. Wang and C.-S. Chien, A spectral-Galerkin continuation method using Chebyshev polynomials for the numerical solutions of the Gross-Pitaevskii equa- tion, J. Comput. Appl. Math., 235 (2011) 2740-2757. [130] Y.-S. Wang and C.-S. Chien, A Two-parameter continuation method for ro- tating two-component Bose-Einstein condensates in optical lattices, Commun. Comput. Phys., 13 (2012) 442-460. [131] J.G. Wang and G.R. Liu, On the optimal shape parameters of radial basis func- tions used for 2-D meshless methods, Comput. Methods Appl. Mech. Engrg., 191 (2002) 2611-2630. [132] H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4 (1995) 389-396. [133] R.A. Williams, S.Al-Assam and C.J. Foot, Observation of vortex nucleation in a rotating two-dimensional lattice of Bose-Einstein condensates, Phys. Rev. Lett., 104 (2010) 050404. [134] G.B. Wright and B. Fornberg, Scattered node compact finite difference-type formulas generated from radial basis functions, J. Comput. Phys., 212 (2006) 99-123. [135] Z. Wu, Compactly supported positive definite radial functions, Adv. Comput. Math., 4 (1995) 283-292. [136] P.S. Yarmolenko, E. J. Moon, C. Landon, A. Manzoor, D. W.Hochman, B.L. Viglianti and M.W. Dewhirst, Thresholds for thermal damage to normal tissues: An update, Int. J. Hyperthermia, 27 (2011) 320-343. [137] R. Zeng and Y. Zhang, Efficiently computing vortex lattices in rapid rotating Bose-Einstein condensates, Comput. Phys. Commun., 180 (2009) 854-860. [138] X. Zhang, K.Z. Zong, M.W. Lu and X. Liu, Meshless methods based on collo- cation with radial basis functions, Comput. Mech., 26 (2000) 333-343. [139] J. Zhou, J.K. Chen and Y. Zhang, Simulation of laser-induced thermotherapy using a dual-reciprocity boundary element model with dynamic tissue properties, IEEE Trans. Biomed. Eng., 57 (2010) 238-245.
摘要: In this dissertation, we study the tailored finite point methods and the radial basis function collocation methods for solving partial differential equations including Schrdinger equation and the bioheat equation.o In the first part, we regard a time-independent Schrdinger equation as a singu-o larly perturbed problem. Under the assumption that the potential is nonnegative and homogeneous with degree greater than zero, we present two theorems for de- scribing the behavior of this problem on an unbounded domain and two conjectures on a bounded domain as the singular parameter approaches zero. We also propose two tailored finite point schemes to obtain the numerical solutions, which agree with the theoretical analysis for the two special cases. Moreover, we exploit an inexact method for the second tailored finite point scheme and compare the performance for various preconditioners with iterative algorithms. In the second part, we study the structure of vortex in a rotating Bose-Einstein condensate trapped by various external potentials by using the radial basis func- tion collocation method (RBFCM) to solve the coupled nonlinear Schrdinger equa-o tions (CNLSE) that describe Bose-Einstein condensate. It is well-known that the accuracy of the radial basis function collocation method is closely related to the shape parameter. At first, we give a rule to select the shape parameter when solving linear Schrdinger equation. Next, we numerically simulate the vortex lat-o tices for isotropic harmonic potentials using the radial basis function collocation method with predictor-corrector (PC) continuation method. Finally, we implement a two-parameter continuation algorithm to observe the dynamical formations of vor- tices under the harmonic potential, harmonic-plus-optical lattice potentials and the harmonic-plus-quartic potentials for various parameters. In the last part, we present an explicit local radial basis function collocation method (LRBFCM) to analyze the temperature distribution during laser heating therapy. Two examples are studied. In the first example we demonstrate the cor- rectness of the proposed method and in the second example, our numerical results show that the local radial basis function collocation method can efficiently solve the three-dimensional bioheat equation and it is easy to implement on complex domains.
在本論文中, 我們研究量身定做的有限點法和徑向基底函數配點法求解偏微分方程式,包括薛丁格方程式和生物傳熱方程式。 在第一部分中, 將不含時薛丁格方程式當作奇異擾動問題。 在位能為非負與齊次, 且次數大於零的假設下, 當奇異參數接近零時, 我們提出兩個定理來描述此問題在無界域上的行為以及兩個在有界域上的猜想。 我們還提出了兩種量身定做的有限點格式以求得此問題之數值解。 對於兩種特殊實例, 我們的數值解與理論分析一致。 此外, 我們運用第二種量身定做的有限點格式的非精確解法並且比較各式預處理元與迭代法的效能。 在第二部分, 我們研究在各種不同外加位能阱中, 旋轉玻色-愛因斯坦凝聚中的渦漩結構。 我們使用徑向基底函數配點法來求得描述玻色-愛因斯坦凝聚的耦合非線性方程之解。眾所周知, 徑向基底函數配點法的準確度和形狀參數有密切關連。 一開始, 我們給予解線性薛丁格方程式時選擇形狀參數的規則。 其次, 我們使用徑向基底函數配點法與預測-修正延續法在數值上模擬同向性簡諧位能阱下之渦漩晶格。最後, 我們實作雙參數延續演算法以觀察在簡諧位能阱, 與各種不同參數之簡諧加光晶格位能阱及簡諧加四次位能阱下旋轉玻色-愛因斯坦凝聚中渦漩的動態形成。 在最後的部分, 我們提出了顯性局部徑向基底函數配點法來分析雷射加熱治療時的溫度分佈。我們研究二個例子。 在第一例中, 我們證明了該方法的正確性而在第二例中, 我們的計算結果顯示, 局部徑向基底函數配點法可以有效求解三維生物傳熱方程式, 並且此方法在複雜域上的問題實作容易。
文章公開時間: 2017-07-16
Appears in Collections:應用數學系所



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